Related papers: Curvilinear High-Order Mimetic Differences that Sa…
Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and…
Mathematical descriptions of flow phenomena usually come in the form of partial differential equations. The differential operators used in these equations may have properties such as symmetry, skew-symmetry, positive or negative…
The treatment of complex geometries in Computational Fluid Dynamics applications is a challenging endeavor, which immersed boundary and cut-cell techniques can significantly simplify by alleviating the meshing process required by…
For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear…
Context. High-resolution numerical methods have been developed for nonlinear, discontinuous problems as they appear in simulations of astrophysical objects. One of the strategies applied is the concept of artificial viscosity. Aims.…
We develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that…
The biharmonic operator plays a central role in a wide array of physical models, notably in elasticity theory and the streamfunction formulation of the Navier-Stokes equations. The need for corresponding numerical simulations has led, in…
Compatible Discrete Operator schemes preserve basic properties of the continuous model at the discrete level. They combine discrete differential operators that discretize exactly topological laws and discrete Hodge operators that…
We discuss a general scheme for a construction of linear conformally invariant differential operators from curved Casimir operators; we then explicitly carry this out for several examples. Apart from demonstrating the efficacy of the…
We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet-to-Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric…
We consider the problem of coarse-graining in the context of finite-volume fluid models. If a variable is defined on a high-resolution grid it may be coarse-grained so that it is defined on a grid of lower resolution. In general this will…
In this paper we shall introduce a simple, effective numerical method for finding differential operators for scalar and vector-valued functions on surfaces. The key idea of our algorithm is to develop an intrinsic and unified way to compute…
We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the…
Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and…
We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model…
We present a mimetic finite-difference approach for solving Maxwell's equations in one and two spatial dimensions. After introducing the governing equations and the classical Finite-Difference Time-Domain (FDTD) method, we describe mimetic…
In this paper, we introduce and discuss an exactly energy-conserving Particle-in-Cell method for arbitrary curvilinear coordinates. The flexibility provided by curvilinear coordinates enables the study of plasmas in complex-shaped domains…
A generalization of implicit conservative numerics to multiple dimensions requires advanced concepts of tensor analysis and differential geometry and hence a more thorough dedication to mathematical fundamentals than maybe expected at first…
Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and / or semiclassical limits of wave equations. We…
In the article, we discuss the conservation laws for the nonlinear Schr\"{o}dinger equation with wave operator under multisymplectic integrator (MI). First, the conservation laws of the continuous equation are presented and one of them is…